Generally, the solid mechanic equations are expressed in a Lagrangian frame, and the fluid part in Eulerian frame. To define and take in account the fluid domain displacement, we use a technique name ALE ( Arbitrary Lagrangian Eulerian ). This allow the flow to follow the fluid-structure interface movements and also permit us to have a different deformation velocity than the fluid one.

Let denote \(\Omega^{t_0}\) the calculation domain, and \(\Omega^t\) the deformed domain at time \(t\). As explain before, we want to conserve the Lagrangian and Eulerian characteristics of each part, and to do this, we introduce \(\mathcal{A}^t\) the ALE map.

This map give us the position of \(x\), a point in the deformed domain at time \(t\) from the position of \(x^*\) in the initial configuration \(\Omega^*\).

Figure 2 : ALE map

\(\mathcal{A}^t\) is a homeomorphism, i.e. a continuous and bijective application we can define as

We study here an incompressible fluid flowing into a cavity, where its walls are elastic. We use the following geometry to represent it.

Figure 1 : Geometry of lid-driven cavity flow.

The domain \(\Omega_f^*\) is define by a square \( [0,1]^2 \), \(\Gamma^{i,*}_f\) and \(\Gamma^{o,*}_f\) are respectively the flow entrance and the flow outlet.
A constant flow velocity, following the \(x\) axis, will be imposed on \(\Gamma_f^{h,*}\) border, while a null flow velocity will be imposed on \(\Gamma_f^{f,*}\). This last point represent also a non-slip condition for the fluid.

Furthermore, we add a structure domain, at the bottom of the fluid one, named \(\Omega_s^*\). It is fixed by his two vertical sides \(\Gamma_s^{f,*}\), and we denote by \(\Gamma_f^{w,*}\) the border which will interact with \(\Omega_f^*\).

During this test, we will observe the displacement of a point \(A\), positioned at \((0;0.5)\), into the \(y\) direction, and compare our results to ones found in other references.

First at all, we can see that the first two tests offer us similar results, despite different orders uses. At contrary, the third result set are better than the others.

The elastic wall thinness, in the stable case, should give an important refinement on the fluid domain, and so a better fluid-structure coupling control. However, the deformed case result are closer to the stable case made measure.